What is the exact permanent residence procedure in Asuncion?

Go to a Paraguayan Embassy website from whatever Paraguayan Embassy is located in a country that speaks your native language.  These web sites will list all the requirements for establishing residency in Paraguay for foreigners.  If your comfortable in English, go to the Paraguayan Embassy in Canada's site.  It has probably the best organized listing of the requirements for residency.   These requirements are constantly changing so to trust a list put together by someone besides an official government site is risky.   As to fee's, the administrative fee's also change regularly but I'm not aware of any official site that lists them.   Be very careful in dealing with "professional helpers" or "lawyers" who promise packaged deals that guarantee getting your residency.  I've noted there's even one group/company who someone wrote about on this forum that is run by people who aren't even based in Paraguay!   The problem with Paraguay now is that so many people are interested in immigrating that there is literally an explosion of shady characters offering their services so be very wary and be advised you must visit Paraguay to start the process.  Although you may leave prior to the process being completed ( 3 to 6 months normally).

Hi, thanks for your reply. As far as I can see from the embassies sites, I need to legalize my birth certificate and police check from the embassy here.
But what are the further steps in  Asuncion? The embassy sites don't give answers, and just refer to their sites.
I believe I do need to apply for a interpol check, medical check, opening a bank account in Asuncion etc. before I can lodge them. I like to know what I have to do, and in what order?
Also what happened if I leave the country before the process is finalized, and return for good to Paraguay after 7 or 8 months?

You do need to open a bank account with the "magic" $5,000 on deposit (you can withdraw the money upon the residency process being complete.   You must also get an interpol check done here in Paraguay at the Policia Nacional as well as getting a medical check done by a local doctor (lab work, doctor exam).   There is no "order" to these items, you just have to get them done prior to turning in your application and paperwork.   As none of these items take just "one visit" to an office to accomplish most people do them all at the same time.  In other words you visit the Policia Nacional to fill out the Interpol paperwork and turn it in, then go back in a couple of days to pick up the paperwork once they do the check and sign off on the paperwork.   Same with doctor, you visit doctor for exam, go to a lab to get your lab work samples taken and then return in a few days to pick up the doctors paperwork.  I think you get the idea.  to the bank to open the account,

As for getting paperwork legalized at a Paraguayan Embassy, I've seen people post here that they no longer have to do that as Paraguay finally accepts an apostle stamp from your home country.   This makes things much quicker.

Once you take care of the items you must be here for (medical check, signing paperwork, etc…) you can leave Paraguay.    Not sure if you can have someone pick up finalized paperwork for you once it's processed or if you must do it in person.  I've seen people say both ways are what must happen.  Most people, to include me and my family, picked everything up in person from Immigration once it was completed.

All these questions can be answered at Immigration in person once you get here.  You'll just need to speak Spanish or have a translator available.  And even if you use a lawyer or "helper" I'd be very sure to get the answers to alll your questions at Immigration before you pay money or leave your application in the sometime capable, and sometimes incompetent, hands of a lawyer or helper.   Good luck !